1. 8 + 7x =2x²
I used my calculator to graph this and got
x is about 4.41 and -.904
2. 1/3x² - 15 = 0
Again, I graphed on my calculator and got x is about 6.70 and -6.70
Are these answers accurate and is there another way to solve?
1. x = (1/4)[7 +/- sqrt(49 + 64)]
= 4.408 or -0.9075
2. (1/3)(x^2 -45)= 0
(x+sqrt45)(x-sqrt45) = 0
x = +/-sqrt(45)= +/- 6.708
To check if the answers obtained from using a calculator are accurate, you can substitute these values back into the original equations and see if they satisfy the equations.
For the first equation, 8 + 7x = 2x²:
Substituting x = 4.41:
8 + 7(4.41) = 2(4.41)²
8 + 30.87 = 38.88
38.87 = 38.88 (rounded)
Substituting x = -0.904:
8 + 7(-0.904) = 2(-0.904)²
8 - 6.328 = 1.633
1.672 = 1.633 (rounded)
Since both substitutions do not result in exactly equal sides of the equation, it means that the values obtained from the calculator may not be entirely accurate.
There are other methods to solve these equations algebraically without relying solely on graphing calculators:
For the first equation, 8 + 7x = 2x²:
Rearrange the equation to the quadratic form: 2x² - 7x - 8 = 0
This can be factored as (2x + 1)(x - 8) = 0
Solving for x, we have two possible solutions:
2x + 1 = 0 -> x = -0.5
x - 8 = 0 -> x = 8
For the second equation, 1/3x² - 15 = 0:
Multiply both sides by 3 to eliminate the fraction:
x² - 45 = 0
Solve for x using the square root property:
x = ±√45
Simplifying, we get:
x ≈ ±6.708
These algebraic solutions give you the exact values of the solutions, rather than just approximations from graphing.